A258651 A(n,k) = n^(k) = k-th arithmetic derivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.
0, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 1, 4, 0, 0, 0, 0, 4, 5, 0, 0, 0, 0, 4, 1, 6, 0, 0, 0, 0, 4, 0, 5, 7, 0, 0, 0, 0, 4, 0, 1, 1, 8, 0, 0, 0, 0, 4, 0, 0, 0, 12, 9, 0, 0, 0, 0, 4, 0, 0, 0, 16, 6, 10, 0, 0, 0, 0, 4, 0, 0, 0, 32, 5, 7, 11, 0, 0, 0, 0, 4, 0, 0, 0, 80, 1, 1, 1, 12
Offset: 0
Examples
Square array A(n,k) begins: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, ... 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, ... 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ... 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, ... 6, 5, 1, 0, 0, 0, 0, 0, 0, 0, ... 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, ... 8, 12, 16, 32, 80, 176, 368, 752, 1520, 3424, ... 9, 6, 5, 1, 0, 0, 0, 0, 0, 0, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..115, flattened
- J. Kovič, The Arithmetic Derivative and Antiderivative, Journal of Integer Sequences 15 (2012), Article 12.3.8
- Wikipedia, Arithmetic derivative
Crossrefs
Columns k=0-10 give: A001477, A003415, A068346, A099306, A258644, A258645, A258646, A258647, A258648, A258649, A258650.
Row 15: A090636, Row 28: A090637, Row 63: A090635, Row 81: A129151, Row 128: A369638, Row 1024: A214800, Row 15625: A129152.
Main diagonal gives A185232.
Antidiagonal sums give A258652.
Cf. also A328383.
Programs
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Maple
d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]): A:= proc(n, k) option remember; `if`(k=0, n, d(A(n, k-1))) end: seq(seq(A(n, h-n), n=0..h), h=0..14);
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Mathematica
d[n_] := n*Sum[i[[2]]/i[[1]], {i, FactorInteger[n]}]; d[0] = d[1] = 0; A[n_, k_] := A[n, k] = If[k == 0, n, d[A[n, k-1]]]; Table[A[n, h-n], {h, 0, 14}, {n, 0, h}] // Flatten (* Jean-François Alcover, Apr 27 2017, translated from Maple *)
Formula
A(n,k) = A003415^k(n).